Finite Group Databasebeta

← #505481

Subgroups — #505481

Order 512.

468 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28013C211yesmaximal, characteristic
2#22809C211yesmaximal, characteristic, complemented
3#13529C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic
5#13507C211yesmaximal, characteristic, complemented
6#13449C211yesmaximal, characteristic
7#13238C211yesmaximal, characteristic
8#13178C211yesmaximal, characteristic, complemented
9#13132C211yesmaximal, characteristic, complemented
10#13131C211yesmaximal, characteristic
11#13131C211yesmaximal, characteristic
12#13105C211yesmaximal, characteristic, complemented
13#13013C211yesmaximal, characteristic
14#12938C211yesmaximal, characteristic, complemented
15#12674C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18#2513—21no—
19#2356—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#346C2211yescharacteristic
33#2073—21no—
34#2065—21no—
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1970C2211yescharacteristic
38#1969C2211yescharacteristic
39#1969C2211yescharacteristic
40#1942C2211yescharacteristic
41#1927—21no—
42#1927—21no—
43#1926—21no—
44#1846C2211yescharacteristic
45#1846C2211yescharacteristic
46#1805—21no—
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4×Q8—21no—
62C2×SD32—22no—
63C2×SD32—22no—
64C2×SD32—41no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C4×Q16—22no—
70C4×Q16—22no—
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#106—21no—
78C2×#106—21no—
79C2×C4⋊C8D411yescharacteristic
80C2×#103—21no—
81C2×#103—21no—
82C2×#103—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#397—41no—
90#397—41no—
91#391—21no—
92#391—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#389—21no—
97#388—41no—
98#388—41no—
99#387—41no—
100#387—41no—
101#383—41no—
102#382—21no—
103#382—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4D411yescharacteristic
108#376—41no—
109#367—22no—
110#367—41no—
111#367—41no—
112#366—22no—
113#366—41no—
114C4⋊C16—22no—
115#358—22no—
116#358—22no—
117#358—41no—
118#357—22no—
119#357—22no—
120#357—41no—
121#346—22no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—21no—
129C22×D4—21no—
130C2×Q16—21no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16#4112yes—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2411yescharacteristic
146C4×Q8—24no—
147C4×Q8—42no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4D811yescharacteristic
157C2×C4⋊C4D811yescharacteristic
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171SD32—44no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16#4112yes—
183C2×C16#4112yes—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107#4112yes—
196#106—81no—
197#106—81no—
198C4⋊C8—41no—
199#103—24no—
200#103—81no—
201#103—81no—
202#102—42no—
203#102—81no—
204#102—81no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—21no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95D811yescharacteristic
224#95D811yescharacteristic
225C24—41no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4C2×D811yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C22×D411yescharacteristic
252Q16—22no—
253Q16—22no—
254Q16—41no—
255Q16—41no—
256D8—41no—
257D8—41no—
258D8—41no—
259D8—41no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8C2×#4111yescharacteristic
270C2×C8C2×#4111yescharacteristic
271C2×C8C2×#4111yescharacteristic
272C2×C8C2×#4111yescharacteristic
273C2×C8H(5,𝔽2)11yescharacteristic
274C2×C82−1+411yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—44no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—161no—
291C22⋊C4—82no—
292C22⋊C4—82no—
293C22⋊C4—161no—
294C22⋊C4—161no—
295C22⋊C4—161no—
296C22⋊C4—161no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—44no—
304C42—82no—
305C42—82no—
306C42—161no—
307C16—22no—
308C16—22no—
309C16—41no—
310C16—41no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23C22×D811yescharacteristic
319Q8—42no—
320Q8—42no—
321Q8—81no—
322Q8—81no—
323D4—81no—
324D4—81no—
325D4—81no—
326D4—81no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4#52511yescharacteristic
367C2×C4#54711yescharacteristic
368C2×C4#55011yescharacteristic
369C2×C4#57511yescharacteristic
370C2×C4#57611yescharacteristic
371C2×C4#57811yescharacteristic
372C8—41no—
373C8—41no—
374C8—41no—
375C8#52911yescharacteristic
376C8#54211yescharacteristic
377C8#55111yescharacteristic
378C8#55211yescharacteristic
379C22—161no—
380C22—161no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22—161no—
385C22—161no—
386C22#310011yescharacteristic
387C22#333411yescharacteristic
388C22#339111yescharacteristic
389C22#341411yescharacteristic
390C22#341411yescharacteristic
391C22#341611yescharacteristic
392C22#341611yescharacteristic
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—82no—
400C4—82no—
401C4—161no—
402C4—161no—
403C4—161no—
404C4—161no—
405C4—161no—
406C4#247911yescharacteristic
407C4#312411yescharacteristic
408C4#334711yescharacteristic
409C4#340211yescharacteristic
410C2—321no—
411C2#2748611yescharacteristic
412C2#3367111yescharacteristic
413C2#3368611yescharacteristic
414C2#3391511yescharacteristic
415C2#3391811yescharacteristic
416C2#3396811yescharacteristic
417C2#3396811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22809C2semidirect
3. #13529C2semidirect
5. #13507C2semidirect
8. #13178C2semidirect
9. #13132C2semidirect
12. #13105C2semidirect
14. #12938C2semidirect
15. #12674C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#280132
#228092
#135292
#135282
#135072
#134492
#132382
#131782
#131322
#131312
#131052
#130132
#129382
#126742